Asymptotically Brownian chain with independent links

1 Asymptotically Brownian chain with independent links 

This chain, of fundamental importance, was studied in the preceding chapter. Its properties can be summarized as follows. The various points of the chain have, vector positions r with j = 0,.. ., N, and the vectors uy 

One shows that, in this case, when N increases, the chain acquires a Brownian behaviour. This behaviour is characterized by the fact that the Fourier transform Pjv(jr) of the probability distribution PN ( ) = 5(r — rN + r0)y is asymptotically given by the simple expression 

Besides, in order to interpret the diffraction experiments made with polymers, one needs the form function 

If the wavelength is very small with respect to the size of the chain, the result becomes even simpler (x 1) 

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Non-asymptotically Brownian chains with independent links chains with intermittence (Levy flights)... 

Polymers appear as chains with excluded volume, but the mathematical difficulties which are encountered when one tries to study these chains have induced many author to use simpler models. These simple chains are either chains with independent links or other asymptotically Brownian chains, and we shall now review these types of chain. 

A chain with independent links is asymptotically Brownian if and only if the mean square length of a link is finite... 

We shall indirectly prove this by exhibiting a whole class of chains with independent links, and a v-Brownian behaviour. Thus, let us associate with each link an independent probability distribution P(u) whose asymptotic behaviour for large u is... 

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